Research projects

Ongoing projects

Coupling FD-DGFEM with RPBEM. Application to Aeroacoustics

In many wave propagation problems the (often unbounded) computational domain is homogeneous. On those cases, the Retarded Potential Boundary Element Method (RPBEM) has been shown to efficient (mainly when coupled with the Fast Multipole Method). In effect, the computations are performed on a surface (on a curve) on 3D (2D) applications, the number of unknowns being much smaller compared to a FD-FEM for example. Moreover, the dispersion error in long time propagation problems is reduced.
In the other hand, in presence of inhomogeneities, the fundamental solution can not be obtained and so the method cannot be applied. In this other situation (heterogeneous and complex media), the Finite Difference (in time) Discontinuous Galerkin Finite Element Method (FD-DGFEM) is a good choice. However the number of degrees of freedom (and in consequence the CPU time) might be very large.
We propose a stable method (via the conservation of a discrete energy) allowing to use the FD-DGFEM on the subdomains where the inhomogeneities are localized, solving in the other regions using RPBEM.

Click on the image to see some numerical results.

Reduced Basis Methods for Wave Propagation Problems

Let us assume that we have an EDP depending on a family of parameters (that can be related, for example, to the physical properties of the model that has been considered or to the geometry of the problem). We consider an output given by the evaluation of a functional over the solution of that PDE for an arbitrary value of the parameters (on a certain range). Our goal is to obtain an accurateapproximation of that output as fast as possible. The reduced basis approximation computes the solution of the parameter dependent PDE as a linear combination of solutions (previously pre-computed) for a given set of samples of the parameters. This method has been recently applied to elliptic equations giving very good results in terms of accuracy and speed of computation. The main objectives are adapting these techniques to wave propagation problems and to analyze the error committed.

Past projects

Higher Order Time Stepping for Second Order hyperbolic Problems and Optimal CFL Conditions

We investigate explicit higher order time discretizations of linear second order hyperbolic problems. We study the even order (2m) schemes obtained by the modified equation method. We show that the corresponding CFL upper bound for the time step remains bounded when the order of the scheme increases. We propose variants of these schemes constructed to optimize the CFL condition. The corresponding optimization problem is analyzed in detail and the analysis results in a specific numerical algorithm. These schemes have been implemented for the scalar wave equation when using a DG higher order method for the space discretization. The numerical rate of convergence are in good agreement with the theoretical ones.

Space-Time Mesh Refinement Methods for Elastodynamics

We are interested in the numericalresolution of the linear elastodynamic equations using explicit schemes in time. The approach that we have followed uses mixed finite element techniques in space and a second order non-dissipative finite differences scheme for the time stepping. In order to take into account the geometrical details and/or the singularities of the solution it's useful to apply local space-time mesh refinement techniques using non-conforming, non-overlapping meshes. The coupling at the interfaces is done using the mortar element technique combined with a discrete energy conservation property that ensures the stability of the scheme since the usual CFL condition is satisfied. The numerical analysis of such schemes is in development.

Click on the image to see some numerical results.

Fictitious Domain Method for Elastodynamics

The model that we have chosen assumes that thenormal stress over the cracks is zero (condition that we call free surface). As the mixed finite element method that we have considered uses regular meshes, we use the fictitious domain method to take into account this homogeneous Neumann boundary condition avoiding the stair-case approximation of the geometry. The boundary condition (the jump of the velocity field through the crack) is imposed in a weak way by means of the introduction of an additional unknown at the physical interface. This variable is discretized using a mesh of the surface. The main advantage of the method is that the mesh used for the volume unknowns does not depend on the geometry. Different finite element spaces are considered. We have been able to show the convergence of the method for the scalar wave equation in a first order formulation. The proof of convergence for elastodynamics is still an open problem due to some difficulties coming from the symmetry of the stress tensor.

It is well known that the solution of elastodynamics is singular on the tips of the cracks. This implies that we will need fine meshes to well capture the behavior of the solution in that region. That's why it is interesting to couple both techniques: space-time mesh refinement methods and the fictitious domain method. When the crack is completely included into the refined region, the algorithm follows straightforward. However, when the artificial interface (associated to the refinement) cross the physical one (associated to the crack), several difficulties raise since the theoretical point of view. We have been able to built two coupling algorithms keeping the same discrete energy conservation property that both methods had separately. The numerical method is then stable. The convergence proof is also an open question.